Generation of VUV, EUV, and X-ray light using VUV-UV-VIS lasers

ABSTRACT

A method for extending and enhancing bright coherent high-order harmonic generation into the VUV-EUV-X-ray regions of the spectrum involves a way of accomplishing phase matching or effective phase matching of extreme upconversion of laser light at high conversion efficiency, approaching 10−3 in some spectral regions, and at significantly higher photon energies in a waveguide geometry, in a self-guiding geometry, a gas cell, or a loosely focusing geometry, containing nonlinear medium. The extension and enhancement of the coherent VUV, EUV, X-ray emission to high photon energies relies on using VUV-UV-VIS lasers of shorter wavelength. This leads to enhancement of macroscopic phase matching parameters due to stronger contribution of linear and nonlinear dispersion of both atoms and ions, combined with a strong microscopic single-atom yield.

This invention was made with government support under grant numbersFA9550-10-1-0561 awarded by the U.S. Air Force; W911NF-13-1-0259 awardedby the U.S. Army Research Office; PHY1068706 awarded by the NationalScience Foundation; PHY1125844 awarded by the National ScienceFoundation; and DE-SC0008803 awarded by the U.S. Dept. of Energy. Thegovernment has certain rights in the invention.

BACKGROUND OF THE INVENTION

U.S. Pat. App. No. 61/873,794, filed 4 Sep. 2013, is incorporated hereinby reference. U.S. Pat. No. 6,151,155, filed 29 Jul. 1998 and issued 21Nov. 2000 is incorporated herein by reference. U.S. Pat. No. 8,462,824,filed 22 Apr. 2010 and issued 11 Jun. 2013 is incorporated herein byreference.

FIELD OF THE INVENTION

The present invention relates to extending and enhancing bright coherenthigh-order harmonic generation into the VUV-EUV-X-ray regions of thespectrum. In particular, the present invention relates to phase matchedand effectively phase matched generation of VUV, EUV, and X-ray lightusing VUV-UV-VIS laser pulses.

DISCUSSION OF RELATED ART

This invention is related to three previously demonstrated techniques,implemented in a novel regime of parameters that contrasts with theparameters of all techniques demonstrated to date:

The first related technique is developed by some of the inventors in thecurrent application and includes a method and device for phase matchingthe generation of high harmonic radiation in a waveguide, which ispatented (U.S. Pat. No. 6,151,155, incorporated herein by reference).Using this technique and, typically, a Ti:Sapphire laser of a wavelengthof 0.8 μm to drive the high harmonic process, full phase matching of theprocess can extend to EUV-soft x-ray wavelengths of about 10 nm.Therefore, in practice, the output flux produced is significant enoughto enable applications using light with wavelength no shorter than 10nm. In this scheme, the generation of soft x-rays of shorter wavelengthsrequires an increase in driving laser intensity. However, scaling ofmacroscopic parameters (see typical values in Table 1, Regime I) doesnot allow for extending the phase matching of the process to shorterx-ray wavelengths, at the same time. Higher intensities result in toomuch ionization of the gas, degrading the phase matching conditions andlimiting both the flux and the coherence of the generated light.Emission at wavelengths shorter than ˜10 nm can be observed but itoriginates from a region of the nonlinear medium of short length, and.orwith a low density of emitters, and in the presence of a large number offree electrons All these factors dramatically decreasing the x-ray fluxto levels not useful for applications. Typically, the emissionoriginates from a short coherence length of <1-100 μm and both atoms andions can emit harmonics of very high photon energies. However, due tothe very short coherence length—that can be much shorter than the mediumabsorption length and medium length—the emission from ions in the highphoton energy range is 3-4 orders of magnitude lower than the emissionfrom atoms limited only within the low photon energy range.

The second technique related to this invention is the idea that alonger-wavelength driving laser can be used to generate X-rays ofshorter wavelength. This idea originates from the simple theoreticalmodel that well describes the high-order harmonic generation process(Kulander; Corkum). However, subsequent theory predicted that usinglonger laser wavelengths λ_(Laser) to generate shorter X-ray wavelengthscomes at a cost of a significantly lower single-atom yield (i.e. theamount of X-ray light generated for each atom in the nonlinear medium,predicted to scales as ˜λ_(Laser) ⁻³). Initial attempts to generateshorter-wavelength X-rays using laser wavelengths as long as 1.5 μm(Shan, PRA 65, 011804 (2001)) supported this prediction of diminishingflux. This and later experiments were done in a gas jet geometry usingmacroscopic parameter regime similar to the well known Regime I (a lowgas pressure (on the order of tens to hundreds of torr) in theinteraction region, and a short propagation length (on the order of0.1-1 mm)), and did not observe flux that would make their approachusable as a general purpose coherent X-ray source. Furthermore, morerecent quantum calculations suggested that the single-atom yield dropseven more dramatically with laser wavelengths and scales as λ_(Laser)^(−5.5). The ideas in this approach imply that shorter-wavelength lasersmay give rise to significantly bright emission from a single emitterlimited to the long-wavelength HHG range, however, the macroscopicoptimization of the HHG upconversion has not been considered neither forshorter-wavelength nor for longer-wavelength lasers.

To remedy the low conversion efficiency observed in the above-mentionedwork, in past work we developed a third technique—a method foroptimization of phase matched conversion of mid-infrared laser lightinto the X-ray region of the spectrum. By using high-pressure medium anda loose-focusing geometry. This approach, devised by many of the sameinventors as the current invention, is described is described inPopmintchev et al., U.S. Pat. No. 8,462,824 (incorporated herein byreference). In contrast with the previously discussed scheme in RegimeI, the inventors demonstrated that both the X-ray radiation from asingle atom, as well as phase matched conversion of this process, can beextended to the generation of much shorter wavelengths. In order toproduce significant X-ray flux, this techniques requires a regime ofparameters (see optimal parameters in Table 1, Regime II), contrastingwith the parameters in the scheme illustrated in Regime I usingTi:Sapphire lasers. This phase matching scaling to shorter wavelengthsis achieved through increasing the laser wavelength while slightlydecreasing the laser intensity. Second, a nonlinear medium of muchhigher density (note the optimal gas pressure of multiple atmospheres)is required when using longer wavelength lasers, to allow the X-ray beamintensity to build to optimum conversion efficiency. This techniqueemploys nonlinear media that is only barely ionized, with many neutralatoms still remaining after the generation process terminates. Carefullydesigned macroscopic parameters lead to coherent build of the X-raysignal over extended optimal medium length that can be a significantlylong (cm or longer). The result is a fully spatially and temporallycoherent, well directed, X-ray beam. Tunability of this broadband X-raysource is achieved by changing the laser wavelength in the IR region andby changing the intensity and/or the medium, plus adjusting the mediumparameters. The tunability range can be from hundreds of nm to sub-nm.Helium gas is a good candidate as a nonlinear medium usable for atunable source over a broad range of X-ray wavelengths. Other noblegases can also be used in some regions of the spectrum. Finally, themethod for optimization in Regime II implies that usingshorter-wavelength laser can generate bright phase-matched HHG frompartially ionized atoms in the long-wavelength VUV range.

A need remains in the art for using shorter wavelength driving lasers inthe VUV-UV-VIS region of the spectrum to benefit from the strongsingle-atom yield, while simultaneously extending this upconversion inan efficient way well into the shorter wavelength VUV, EUV, and X-rayregions.

SUMMARY OF THE INVENTION

It is an object of the present invention to use shorter wavelengthdriving lasers in the VUV-UV-VIS region of the spectrum for phasematched generation and effectively phase matched generation (definedherein as generation with optimized phase mismatch) of coherent VUV,EUV, and X-ray light.

Embodiments extend and enhance bright coherent high-order harmonicgeneration into the VUV-EUV-X-ray regions of the spectrum. This methodinvolves a previously-unrealized, straightforward way of achievingrelatively high upconversion of laser light to VUV, EUV, and X-ray lightat conversion efficiency that in some cases can approach 10⁻³, Thisextension of HHG relies on using VUV-UV-VIS lasers of shorter wavelengthto drive the upconversion process, and employing focused intensities ofthese lasers that cause substantial ionization of the nonlinear medium.We show that the use of short-wavelength driving lasers in a properlydesigned loose-focus, waveguide geometry, or laser self-guiding geometryover an extended distance that is similar of greater than a coherencelength can unexpectedly lead to a non-obvious, favorable enhancement ofmacroscopic phase matching parameters due to stronger contribution oflinear and nonlinear dispersion of both atoms and multiply charged ions.The resulting strong macroscopic yield of VUV-to-EUV-to-X-ray photons iscombined with the significantly increased microscopic single-atom yieldthat is inherent to driving HHG with shorter wavelength lasers. In aproperly designed and implemented geometry, well-directed, ultrafastVUV-EUV-X-ray beams of full temporal and spatial coherence can exhibit anearly quasi-monochromatic spectrum that consists of a relatively smallnumber of well-separated harmonics, or a single quasi-monochromaticharmonic. In the time domain, the high harmonic light can be an isolatedpulse, a train of pulses whose separation decreases as we decrease thedriving wavelength, or this train of pulses may even merge into a singlerelatively long-duration pulse that corresponds to an isolated singleharmonic. The generated harmonics are also semi-continuously tunablethrough varying the laser wavelength and/or changing the nonlinearmedium.

Embodiments of the present invention use, first, shorter wavelengthdriving lasers in the VUV-UV-VIS region of the spectrum (see Regime IIIin Table 1), in contrast to the work of U.S. Pat. No. 8,462,824. Second,simultaneously with decreasing the laser wavelength the presentinvention significantly increases the laser intensity by few orders ofmagnitude to the range of 10¹⁵-10¹⁶ W/cm² or greater. A fundamentallynew regime of HHG can be accessed when laser wavelength, peak power, andfocusing geometry are properly optimized. The illumination at thisintensity results in relatively strong ionization of the nonlinearmedium, often creating multiply charged ions (e.g. Ar⁺, Ar²⁺, Ar³⁺,Ar⁴⁺, Ar⁵⁺, etc). Furthermore, using shorter VUV-UV-VIS laser means thatboth the linear and nonlinear index of refraction of the neutral atomicand ionic nonlinear medium are significantly large since these laserwavelengths approach the UV resonances of atoms. Finally, we extend theinteraction length with the nonlinear medium where such high laserintensities are maintained by using loose-focusing, waveguidinggeometry, or laser self-guiding geometry to a length that is optimal forefficient HHG upconversion.

The present invention is generally separated into two regimes: RegimeIIIA of high laser intensity where full phase matching balance can beachieved, and Regime IIIB where there is finite phase mismatch but it isstrongly mitigated.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a schematic block diagram illustrating apparatus forphase-matched and effectively phase-matched high-order harmonicgeneration (HHG) in the VUV-EUV-X-ray regions of the spectrum accordingto the present invention.

FIG. 1B is schematic block diagram showing injecting of a nonlinearmedium in the interaction region and optional differential pumping tovacuum.

FIG. 2A is a schematic block diagram illustrating a guided beam geometryembodiment of the present invention.

FIG. 2B is a schematic block diagram illustrating a loose-focusinggeometry embodiment of the present invention.

FIG. 2C is a schematic block diagram illustrating a laser self-guidinggeometry embodiment of the present invention.

FIG. 3A is an isometric schematic block diagram illustrating interactionwithin a waveguide for strong, constructive, coherent build up of HHGlight.

FIG. 3B is a side-view schematic block diagram illustrating how outputpulse intensity increases and beam diverge decreases with phase matchingor effective phase matching within the medium.

FIG. 3C is a side-view schematic block diagram illustrating coherencelength within the medium according to the present invention.

FIG. 3D is a plot showing the intensity of the output pulses with phasematching over the interaction region length comparable to a shortabsorption length.

FIG. 3E is a plot showing the intensity of the output pulses witheffective phase matching over the interaction region comparable to along absorption length.

FIG. 4A is schematic flow diagram illustrating phase velocity matchingof the driving laser light and the generated light in Regime IIIAaccording to the present invention.

FIG. 4B is a schematic flow diagram showing the reduced phase velocitymismatch in Regime IIIB in the present invention.

FIG. 4C (Prior Art) is a schematic flow diagram showing phase velocitymatching with Regime II.

FIG. 5A is a schematic flow diagram illustrating resulting minimal groupvelocity mismatch of the driving laser light and the generated light inRegime IIIA according to the present invention.

FIG. 5B is schematic flow diagram illustrating resulting reduced groupvelocity mismatch in Regime IIIB according to the present invention.

FIG. 5C (Prior Art) is a schematic flow diagram illustrating resultingstrong group velocity mismatch of the driving laser light and thegenerated light in prior art Regime II.

FIG. 6A is a detailed schematic flow diagram illustrating the phasevelocity matching of the driving laser light and the generated lightwhere dispersion of atoms and of low charged ions can compensate largeplasma dispersion in Regime IIIA according to the present invention.

FIG. 6B is a detailed schematic flow diagram illustrating the reducedphase velocity mismatch where dispersion of atoms and predominantlydispersion of multiply charged ions can mitigate larger plasmadispersion in Regime IIIB according to the present invention.

FIG. 6C (Prior Art) is a detailed schematic flow diagram illustratingthe phase velocity matching of the driving laser light and the generatedlight in prior art Regime II, where atomic dispersion compensates plasmadispersion and ionic dispersion is negligible.

FIG. 7A (Prior Art) is a schematic spectral domain diagram illustratingthe regions of the laser spectra and the upconverted HHG spectra withphase matching according to prior art Regimes I and II.

FIG. 7B is a schematic spectral domain diagram illustrating the regionsof the laser spectra and the efficient upconverted HHG spectra accordingto present invention Regimes IIIA and IIIB.

FIG. 8 is a schematic spectral domain diagram illustrating output pulsespectral shape of upconverted HHG light according to prior art Regime IIand the present invention, Regimes IIIA and IIIB.

FIG. 9A (Prior Art) is a schematic diagram illustrating the controlparameters, optimizing parameters, and resulting macroscopic HHG flux inconventional for prior art Regime I, and shows parameters and results atvarious wavelengths.

FIG. 9B (Prior Art) is a schematic diagram illustrating the controlparameters, optimizing parameters, and resulting macroscopic HHG flux inconventional for prior art Regime II, and shows parameters and resultsat various wavelengths.

FIG. 9C is a schematic diagram illustrating the control parameters,optimizing parameters, and resulting macroscopic HHG flux Regime IIIaccording to the present invention.

FIG. 10A (Prior Art) is a schematic time domain diagram illustrating theregions of the driving laser pulse durations and the HHG pulse durationswith phase matching in prior art Regimes I and II at variouswavelengths.

FIG. 10B is a schematic time domain diagram illustrating the regions ofthe driving laser pulse durations and the HHG pulse durations with phasematching in prior art Regime I and in phase matching and effective phasematching in Regime III according to the present invention at variouswavelengths.

FIGS. 11A-D are time domain diagrams illustrating input laser pulseshapes and output HHG pulse shapes according to prior art Regime II.

FIGS. 12A-D are time domain diagrams illustrating input laser pulseshapes and output HHG pulse shapes according Regime III of the currentinvention.

FIG. 13 is a time domain diagram presenting the generated, chirped, HHGpulse durations and the minimum HHG pulse durations if the pulses arecompressed for prior art Regime II and Regimes IIIA and IIIB accordingto the present invention at various wavelengths.

FIG. 14 is plot of HHG phase matching limits for prior art Regime II andefficient HHG upconversion photon energies in Regimes IIIA and IIIBaccording to the present invention.

FIG. 15A is a plot showing example experimental data demonstratingspectrally selected bright discrete harmonics in the X-ray region of thespectrum up to the water window in Regime IIIB according to the presentinvention.

FIG. 15B is a plot showing example experimental data presenting brightdiscrete harmonics spanning the VUV-EUV-X-ray spectral regions andcorresponding multiply charged emitters.

FIG. 16 is a plot showing example experimental data demonstratingultra-bright discrete harmonics in the EUV spectral region in RegimeIIIA according to the present invention.

FIG. 17 is a plot showing an example calculation of the criticalionization level of a gas versus laser wavelength for Regime IIIAaccording to the present invention, with and without spatial averaging.

FIG. 18A is a plot showing an example calculation of the expected HHGemission from a single atom versus driving laser wavelength for RegimeIIIA according to the present invention.

FIG. 18B is a plot showing an example calculation of the expectedefficient macroscopic harmonic intensity versus corresponding HHG photonenergy and driving laser wavelength for Regime IIIA according to thepresent invention.

FIG. 19A is a plot showing example experimental data demonstrating beamprofile and a bright narrow-band EUV harmonic around 13 nm isolated by acombination of low-pass and high pass thin film filters and by using aspecific laser intensity.

FIG. 19B is a plot showing example experimental data of two brightnarrow-band EUV harmonics around 13 nm isolated by a combination oflow-pass and high pass thin film filters and by using a specific laserintensity.

FIG. 19C is a plot showing example experimental data demonstrating abright narrow-band VUV harmonic isolated by a single low-pass and highpass thin film filter and by using a specific laser intensity.

FIG. 20A is a plot showing an example of an extended temporal window ofefficient upconversion over a large number of driving laser cycles forRegime III according to the present invention.

FIG. 20B (Prior Art) is a plot showing an example of a narrow temporalwindow of upconversion for prior art Regime I.

FIG. 21 is a flow diagram illustrating steps in the process of globaloptimization of HHG with phase matching or effective phase matchingaccording to the present invention, and steps in selecting the optimalparameters for most efficient HHG at a desired HHG wavelength.

FIG. 22 is a plot showing example experimental data demonstratingpressure optimization of UV-driven HHG spanning the EUV-VUV-X-rayregions of the spectrum for Regime IIIB according to the presentinvention.

DETAILED DESCRIPTION OF THE INVENTION

Prior to this invention it was widely accepted that phase matching inextreme frequency upconversion in highly ionized plasma couldn't beobtained due to the fact that free electron dispersion dominates theprocess and no longer is balanced by the neutral atom dispersion. Here,we demonstrate that it is possible to achieve such balance or stronglymitigate the phase mismatch.

Using shorter wavelength VUV-UV-VIS lasers to drive the HHG processmeans that both the linear and nonlinear index of refraction of theneutral atomic nonlinear medium are significant since these laserwavelengths approach the UV resonances of atoms. As a result, theoptimal laser intensity is significantly higher which further increasesthe nonlinear dispersion contribution which increases with laserintensity ˜n₂ ^((atom))*I_(L)(time). Next, linear and nonlinear indicesof refraction of ions can be nearly as large as those of neutral atoms,and will increase as we use shorter driving laser wavelengths.Furthermore, since highly charged ions are created later in the laserpulse when the laser intensity is higher, the nonlinear dispersion termcan be also significant because of the higher laser intensity ˜n₂^((ion))*I_(L)(time). Thus there is significant dispersion contributionboth from neutral atoms and ions. This is balanced against a freeelectron dispersion term of opposite sign, and of greatly diminishedmagnitude compared with its value at longer infrared wavelengths.Furthermore, the coherence length of the upconversion process to firstapproximation (assuming dominant plasma dispersion) scales approximatelyas L_(coh)˜1/(q*I_(Laser)) with q being the harmonic order, and whichfor a given X-ray energy is smaller if a shorter-wavelength driver isused. Combined, these macroscopic propagation factors allow forsignificant coherent signal buildup for X-ray HHG even in the case ofvery high laser intensity resulting in very high limiting ionizationlevels that in some cases can be >100%.

When the length of the nonlinear medium, the absorption depth of themedium to the generated X-rays, and the coherence length all reachcomparable values, HHG upconversion is effectively phase matched. Boththese scaling considerations as well as experimental data suggest thatoptimization of the interaction geometry can approach this ideal in thecase of short-wavelength driving lasers, and very high intensity andcorresponding very high photon energies.

Furthermore, as mentioned previously the X-ray emission yield per atomalso increases strongly with the use of shorter drive laser wavelengths.This results in shorter excursion distance of the rescattering electron.Due to lower quantum diffusion, the recombination probability issignificantly increased, with yield from a single atom that scalesstronger than ˜I_(Laser) ^(−5.5)·TDSE calculations show that thesingle-atom yield under phase matched illumination scales as ˜I_(Laser)^(−7.5) or even faster since much higher laser intensities compared toRegime II are allowed. Thus, regardless of having a finite coherencelength the conversion efficiency can be significantly high compared tothe prior art.

Table 1 shows a comparison of practical parameters when scaling the HHGprocess towards generating shorter X-ray wavelengths, using near-IR 0.8mm driving laser (Regime I—prior art), longer-wavelength mid-IR drivinglasers (Regime II—prior art), and shorter-wavelength VUV-UV-VIS lasers(Regime III—present invention).

TABLE 1 Regime I (prior art) Regime II Regime III (non-phase- (priorart) (present invention) matched) (phase-matched) (phase-matched) I_(L)= 0.8 mm = I_(L) increases & I_(L) decreases & const & I_(L) decreasesI_(L) increases I_(L) increases (slightly) (significantly) Peak LaserIncreases Decreases from Increases Intensity significantly 10¹⁴-10¹⁵ tosignificantly [W/cm²] from 10¹⁴ under phase from 10¹⁴ 10¹³-10¹⁴ matchingto >10¹⁶-10¹⁷ to >10¹⁶. conditions. under phase matching conditions oreffectively phase matching conditions. Ionization Increases Low andIncreases Level of from ~10% decreases from ~10% the Medium to >>200%from ~10% to >>600% Highly ionized or to ~0.001%. Highly ionized ormultiply-ionized Gently ionized multiply-ionized nonlinear nonlinearnonlinear medium. medium. medium. Emitters of Ions generated Atomsgently Highly charged coherent by the driving ionized by the ions (ionstates light laser or pre- driving laser. of +1 to >+6) ionized medium.Emission from generated by the Poor emission atoms only. driving laseror from ions. using pre-ionized medium. Very strong emission from bothatoms and ions. Nonlinear Decreases Increases Increases Medium from~10-500 from ~10-500 from ~1-10 Density torr to ~1 torr to >>10000 torrto >1000 torr. torr. torr. Range: fraction Range: multi- Range: fractionof atmosphere. atmosphere. of atmosphere to >atmosphere. Index of Lowlinear and Very low linear Very large linear refraction nonlinear indexand nonlinear and nonlinear of the medi- of refraction per index ofindex of um at the neutral atom or refraction per refraction per drivinglaser ion. neutral atom. neutral atom and wavelength ion. UsefulDecreases from Increases from Increases from Nonlinear mm-cm to mm.mm-cm to m. mm to cm Medium Length Density of Increases Constant: ~>10¹⁷Increases the Ionized from >10¹⁷ cm⁻³. from >10¹⁷ Electrons to >>10¹⁸cm⁻³. to >>10¹⁸ cm⁻³. Conversion 10⁻⁵ to <10⁻¹¹ 10⁻⁵ to <10⁻⁷- 10⁻³ to<10⁻⁵- efficiency 10⁻⁸ 10⁻⁶ Phase Plasma Neutral atom Large neutralmatching dispersion dispersion atom + dispersion dominates - no balanceslow low/multiply- balance dispersion plasma charged ion balance.dispersion. dispersions counteract large plasma dispersion. VUV-EUV-Rapidly Same or Comparable flux X-ray decreasing at increasing at fromatoms and Flux shorter X-ray shorter X-ray ions at longer wavelengths.wavelengths. VUV-EUV and Non-phase Phase-matched shorter X-ray matchedaddition of X-ray wavelengths. addition of X-ray fields. Phase matchedfields. and effectively phase matched addition of VUV/EUV/X-ray fieldsor very long coherence length compared to absorption length/mediumlength.

FIG. 1 is a schematic block diagram illustrating apparatus forphase-matched and effectively phase-matched high-order harmonicgeneration into the VUV-EUV-X-ray regions of the spectrum according tothe present invention.

The embodiment of FIG. 1 might utilize a driving laser source 102providing, for example, a 0.4 μm pulse, a 0.27 μm pulse or a 0.2 μmpulse 102 with a peak intensity of >3×10¹⁴ W/cm² and can exceed 10¹⁵W/cm²-10¹⁶ W/cm². For example source 102 might be a laser or a chirpedpulse amplifier (CPA) directly producing light at the desired drivingwavelength 104, a laser in conjunction with an optical parametricamplifier (OPA) for converting the laser light to the desired drivingpulse 104 wavelength, or an optical parametric chirped pulse amplifier(OPCPA), or an excimer laser, or a harmonic beam of a Ti:Sapp laser orof a Ti:Sapp CPA/OPCPA or other solid-state laser or of OPA or OPCPAsystem (i.e. a 2nd, 3rd, 4th, 5th, etc. harmonic that is at the desiredwavelength 104), or for example a VUV-UV-VIS free electron laser (FEL)or a harmonic of an IR FEL. The driving laser might be also a laser beamgenerated through frequency mixing of two or more laser beams ofdifferent frequency in a waveguide or a gas cell. The nonlinear medium120 is highly ionized, for example from about 1-10% to above 600%. Thepressure of the medium can be 10-1000 torr to multi-atmosphere. Thedensity of the ionized electrons can be >10¹⁷ to >>10¹⁸ cm⁻³. Thisresults in a useful nonlinear medium length L_(medium) of centimeters tosub 10-100 mm. The phase-matched addition of the HHG fields means thatflux does not vary strongly with wavelength.

The laser source 102 energy, wavelength, and pulse 104 duration areselected to maintain phase matched or effectively phase matched HHG 106generation. Driving source 102 may produce ultrashort driving pulses atany repetition rate or long “macroscopic” pulses at any repetition ratewith multiple driving pulses under the envelope.

Medium 130 or 132 might comprise atomic gases (for example, noble gases:helium, neon, argon, krypton, xenon, etc.) and mixtures, of moleculargasses (for example hydrogen gas) and mixtures of atomic and moleculargases (for example air, etc.). Gas mixing apparatus 134 is optional. Inmixtures, phase matching and effective phase matching relies on thepresence of a target that is less or non-ionized compared to the otherspecies targets for a given peak laser intensity. Therefore, a mixtureof targets with different ionization potentials may be desirable.Mixtures allow the less ionized medium to contribute to the neutral andionic index of refraction. Therefore, higher laser intensity can beemployed. As a result higher photon energies may be phase matched oreffectively phase matched with further increase in HHG flux. Since theshort wavelength driving laser can penetrate higher density plasma, thenonlinear medium may be liquid, or mixture of liquids, or for examplesolid state He, Ne, etc, such that at the maximum ionization level thecritical plasma density at the driving laser wavelength is not exceeded(for example 50% of the critical plasma density is reached at a meancharge Z=1 for H₂O and 0.16 mm laser, 67% of the critical plasma densityis reached at a mean charge Z=1 for SiO₂ and 0.16 mm laser; 59% of thecritical plasma density is reached at a mean charge Z=1 for solid H₂ and0.16 mm laser).

Other optional features include a front mirror 122 for beam steering atthe input to monitor beam diameter at the entrance of waveguide or gascell 120. Laser beam 136 is coupled into monitoring equipment 138 inorder to optimize waveguide mode coupling, to monitor and control beampointing, as well as to monitor the laser peak intensity on target.Monitoring 138 may also monitor the input spectrum. Front mirror 122 maybe removable.

Optional back mirror 112 selects beam 114 for monitoring equipment 116.Monitoring equipment 116 may image the beam diameter at the output or atcertain propagation distance, may monitor and control beam pointing, maymonitor the beam intensity at the output, and/or may monitor the outputlaser spectrum. Front mirror 112 may be removable or have a hole fortransmission.

Optional differential pumping 142, 144 or several of them may beprovided at input and output of waveguide or gas cell 120, asillustrated in FIG. 1B. In this embodiment, small diameter tubing isprovided along the axis of the laser and HHG propagation, and the tubingrestricts the gas flow for an embodiment where the nonlinear medium isin a gas phase, while allowing the driving laser and the generated lightthrough. The gas flow resistance increases significantly with decreasesin the diameter of the tubing and scales as ˜D⁻⁴, where D is thediameter. The gas flow coming out of the region containing the nonlinearmedium can preferentially be redirected towards a bigger diameter regionof low resistance pumped by vacuum pumps and can be also additionallysteered in this direction by inclined surfaces (e.g. 45 degrees in theexample drawing).

FIG. 2A is a schematic block diagram illustrating a guided beam geometryembodiment of the present invention. This embodiment comprises a hollowwaveguide (cylindrical or tapered) filled with a nonlinear medium. Thelaser beam 104 is focused to an optimal beam diameter relative to aselected waveguide diameter to couple to a desired waveguide mode or tominimize attenuation losses. The idea is to extend the region of highlaser intensity, i.e. the Rayleigh range z_(Rayleigh), towards multipleRayleigh ranges. This is of a significant importance to decrease thetechnical challenges with respect to the required laser technologyhaving in mind that the optimal high laser intensities in some casesexceeds >10¹⁵-10¹⁶ W/cm². Also, this geometry can provide near planewave propagation of the laser beam. Such uniform conditions may allowfor a better collinear (effectively) phase-matched upconversion. Thewaveguide can also provide means to contain a gas with uniform pressureover extended distances and can also provide pressure-length productsthat are uniform or with predesigned gradients that are bettercontrolled compared to a gas directly injected into a vacuum chamber.

FIG. 2B is a schematic block diagram illustrating a loose-focusinggeometry embodiment of the present invention. In a loose-focus geometrythe medium length can be much shorter than the region of high laserintensity, the Rayleigh range z_(Rayleigh). Thus, the interaction regionis almost in a near plane wave propagation similar to propagation in awaveguide. The required optimal laser energy can be significantlylarger.

FIG. 2C is a schematic block diagram illustrating a laser self-guidinggeometry embodiment of the present invention. A waveguiding geometry mayalso assist the formation of self-guided propagation of the laser beam.The laser beam can collapse to smaller diameter than that at theentrance of the waveguide and can propagate steadily after a certaindistance, or can propagate at the same diameter regardless ofplasma-induced defocusing. This is due to a balance between Kerr lensself-focusing and plasma-induced defocusing specific for a waveguidestructure (i.e. different from a conventional filamentation effect infree space) and may coincide with a regime of nonlinear mediumparameters and laser pulse parameters that are also optimal forefficient HHG upconversion. Such self-guiding in space, a stablepropagation in space, may be also accompanied by a stable propagation ofthe laser pulse in time without any change in pulse duration or with acompression in pulse duration along the interaction distance.

FIGS. 3A-3D are schematic diagrams illustrating how the optimizedVUV-EUV-X-ray intensity increases within the medium, as well as theparameters of the HHG scheme that depend on the driving laser wavelengthand the HHG wavelength.

FIG. 3A is an isometric schematic block diagram illustrating interactionwithin a waveguide for a phase matched and effectively phase matchedbuildup of high-harmonic signal. FIG. 3A also shows the phase velocityand the resulting group velocity of the driving laser pulse associatedwith the propagation of the carrier wave and its envelope, respectively.In an ideal case, the phase velocities of the driving light and thegenerated light has to be matched while the group velocity mismatch hasto be mitigated. FIG. 3A also shows interactions within the medium inthe waveguide, wherein the harmonic fields are generated, and are addedconstructively over a finite interaction length.

FIG. 3B is a side-view schematic block diagram illustrating how outputpulse intensity increases and the HHG beam divergence decreases with(effective) phase matching within the medium. The driving pulse phasevelocity is matched to the phase velocity of the generated HHG or themismatch is greatly minimized. Thus, the coherence length of the(effective) phase matching is extended beyond the optimal medium lengthL_(medium) in FIG. 3C (ideally L_(coh)→∞ for some conditions, or L_(coh)can be finite)—in other words, the entire interaction region is nowequivalent to a (+) region of constructive addition of HHG fields.

FIG. 3C is a side-view schematic block diagram illustrating coherencelength within the medium according to the present invention. In FIG. 3Cconstructive addition of HHG fields is possible, in contrast to a (−)zone where destructive addition of HHG fields can greatly reduce the HHGsignal. No significant destructive interference occurs. If there isfinite phase mismatch such a selection of the coherence length with therespect of the medium length can be denoted also as an effective phasematching. In the present art, the coherence length L_(coh)(z,t, I_(L),I_(HHG)) for any HHG wavelength can be maintained always relatively longby properly selecting the laser parameters and the nonlinear mediumparameters.

The coherence length of the upconversion process to first approximationscales as L_(coh)˜1/(q*I_(Laser)), assuming very high level ofionization, and the relatively low harmonic orders correspond to highphoton energies when we use short wavelength lasers. As a result, thisapproach allows for significantly long coherence length, that may becomparable to the optimal length of the medium (i.e. close to one or fewabsorption lengths of the medium), even for high photon energies.Selecting the optimal medium length to be comparable to the relativelylong coherence length ensures strong constructive buildup of harmonicsignal even at very high ionization levels/strong free electrondispersion.

FIG. 3D is a plot showing the resulting intensity of the output HHGpulses over the interaction region. It increases strongly withinteraction length, until absorption of the generated HHG or evolutionof the laser pulse in the gas limits further buildup of HHG signal.Under phase matching conditions or greatly reduced phase mismatch, thereis coherent radiation build-up in a very specific direction that reducesthe HHG beam divergence, as shown in FIG. 3B.

FIG. 3E is a plot showing that effective phase matching may be achievedalso for a finite coherence length that can be relatively short (e.g.1-500 mm), however, the medium length can be selected to be shorter butcomparable. Under such conditions a strong coherent buildup can still beachieved and a desired number of photons can be obtained since theemission from a single atom significantly increases for shorter laserwavelengths. In such an optimization, the absorption medium length maynot be reached and absorption limited HHG upconversion may not beobtained.

FIGS. 4A-6C illustrate phased matched HHG in three regimes: prior artRegime II, Regime IIIA according to the present invention, and RegimeIIIB according to the present invention. Refer back to Table 1 for thecharacteristics of each Regime. Briefly, in Regime II, phase matching isachieved as a balance between neutral atom dispersion and free electronplasma dispersion that has an opposite sign in a lightly ionized medium.In Regime IIIA, the balance is between the dispersion of neutral atomsin combination with the dispersion of low-charged ions of the same sign,and the plasma dispersion in a highly ionized medium. The resultinggroup velocity mismatch here is minimal. In Regime IIIB, full balancecannot be achieved, but the phase mismatch is minimized by balancingbetween dispersion of neutral atoms plus multiply-charged ions andplasma dispersion in a multiply-charged medium. The resulting groupvelocity mismatch here increases but it can be relatively small comparedto that in Regime II.

FIG. 4A is a schematic flow diagram illustrating phase velocity matchingof the driving laser light and the generated light in Regime IIIAaccording to the present invention. In Regime IIIA, phase matching isoptimized in a highly ionized medium. To optimize macroscopic phasematching in the process of HHG means that the driving laser pulse andthe generated HHG pulses propagate through the nonlinear medium almostat the same phase velocity. When HHG emission is in the EUV and HHGregion of the spectrum the HHG phase velocity is very close the speed oflight in vacuum v_(HHG)≈c and may vary insignificantly when changing theparameters of the medium. Therefore, the phase velocity of the drivinglaser is the one that has to be primarily optimized through the laserand medium parameters and has to be also v_(LASER)≈c. In principal,while the plasma dispersion speeds up the phase velocity of the laser,the normal neutral atom and ionic dispersions slow it down. Since almostall harmonics propagate at the same phase velocity, obtaining phasematching in HHG upconversion usually results in simultaneous phasematching for the whole HHG bandwidth, which can be the case in RegimesIIIA and B and Regime II. Note that for HHG in the VUV region thedeviation from c may be significantly bigger and the optimal parametersmay be modified in comparison to HHG in the EUV and X-ray regions.

In Regime IIIA, the indices of refraction (both linear and nonlinear) ofboth atoms and ions increase as we decrease the laser wavelength towardsthe VIS-UV-VUV regions. Furthermore, in this regime the indices ofrefraction of ions are comparable to that of atoms. Thus, under someconditions much larger plasma dispersion can be balanced by neutral atomand ionic dispersions (see FIG. 6A). It has been considered in the pastthat such a balance is not possible since plasma dispersion dominatesthe interaction leading to a significant phase mismatch and poorupconversion efficiency. In Regime IIIA the balance may be achievedbetween neutral atom including low charged ionic dispersion andrelatively high plasma dispersion. In some instances the neutral atomsmay be ionized and not present later in the laser pulse.

FIG. 4B is a schematic flow diagram showing the reduced phase velocitymismatch in Regime IIIB in the present invention. At very high laserintensities in Regime IIIB, multiply charged ions can be generated.Under such conditions full balance between the phase velocities cannotbe achieved, however, the phase mismatch is relatively low allowing forefficient upconversion. Since the plasma dispersion is larger at largerionization levels (due to higher laser intensity), although the totalindex of refraction (including linear and nonlinear indices) increasesof both neutral atoms and ions, the phase velocity of the laser isslightly greater than but close to c, v_(LASER˜)C (see FIG. 6B). InRegime III B the neutral atoms may disappear completely quite earlyduring the laser pulse and most of the higher order harmonics may beefficiently generated predominantly only from ions. In some cases, asmany as few multiply charged ions can counteract the much higher plasmadispersion to greatly reduce the phase mismatch or achieve effectivephase matching over a properly selected medium length.

FIG. 4C (Prior Art) is a schematic flow diagram showing phase velocitymatching with Regime II. In prior art Regime II, phase matching isobtained as a balance between neutral atom dispersion and free electronplasma (see FIG. 6C). The balance in Regime II can be achieved only upto relatively low plasma dispersion since the indices of refraction ofatoms and ions decrease and are relatively low as we increase the laserwavelength towards the mid-IR. In this regime, the contribution of anyionic dispersion is negligible.

FIG. 5A is a schematic flow diagram illustrating resulting groupvelocity matching of the driving laser light and the generated light inRegime IIIA according to the present invention.

While the phase velocity of the driving laser pulse is associated withthe propagation of the carrier frequency of the electric field, thegroup velocity is associated with the propagation of the envelope of thefield (refer to FIG. 3A). In principal, for a medium with normaldispersion the group velocity is always smaller than the speed of lightin vacuum V^(GROUP) _(LASER)<c and full group velocity matching cannotbe achieved. Most of the HHG light propagates not only with a phasevelocity but also with group velocity very close to c, V^(GROUP)_(HHG)≈c.

In Regime IIIA, the resulting group velocity mismatch under optimalconditions is significantly minimized when the laser wavelength isdecreased. This means that a UV driving pulse with either a smaller or alarger number of cycles can maintain phase matching and one can generateeither an isolated HHG burst or a train of burst with similar efficiencyper laser cycle. In contrast, for a few-cycle mid-IR driving pulse phasematching can be destroyed for a relatively small density-length productand efficient upconvertion can be precluded. This is because theenvelope of the laser field lags with respect to the carrier field andthe amplitude of the carrier is modified, hence, phase matching isaffected. Thus, unless any additional technique is used, a larger numberof cycles may be desirable for a mid-IR laser pulse to obtain groupvelocity mismatch that is effectively reduced in order to preserve phasematching.

FIG. 5B is schematic flow diagram illustrating resulting reduced groupvelocity mismatch in Regime IIIB according to the present invention. Inthe present invention in Regime IIIB, at much higher laser intensitiesand higher gas pressures compared to Regime IIIA, the group velocitymismatch increases. However, since the optimal propagation length can berelatively short the group velocity walk off is effectively smaller.

FIG. 5C (Prior Art) is a schematic flow diagram illustrating resultinggroup velocity mismatch of the driving laser light and the generatedlight in prior art Regime II. When the phase velocities are matched inthe conventional Regime II, as we increase the laser wavelength there isa resulting group velocity mismatch that is severe. This is due to highoptimal pressure-length products required to achieve phase matching andto compensate for low emission from a single atom. Additionaloptimization (e.g. increasing the number of laser cycles, pulse shaping,etc.) or techniques are needed to mitigate the effect and preserve phasematching.

FIG. 6A is a detailed schematic flow diagram illustrating the phasevelocity matching of the driving laser light and the generated lightwhere dispersion of atoms and low charged ions can compensate largeplasma dispersion in Regime IIIA according to the present invention. Asdiscussed with respect to FIG. 4A, phase matching is achieved as adispersion balance between atoms and low-charged ions in a highlyionized medium.

FIG. 6B is a detailed schematic flow diagram illustrating the reducedphase velocity mismatch where dispersion of atoms and multiply chargedions can mitigate larger plasma dispersion in Regime IIIB according tothe present invention. As discussed with respect to FIG. 4B, fullbalance cannot be achieved, but the phase mismatch is minimized bycounterbalancing dispersion of atoms and multiply-charged ions and veryhigh plasma dispersion in a multiply-charged medium.

FIG. 6C (Prior Art) is a detailed schematic flow diagram illustratingthe phase velocity matching of the driving laser light and the generatedlight in prior art Regime II, where ionic dispersion is negligible. Asdiscussed with respect to FIG. 4C (Prior Art), phase matching isachieved as a balance between neutral atom dispersion and free electronplasma dispersion in a lightly ionized medium.

FIG. 7A (Prior Art) is a schematic spectral domain diagram illustratingthe regions of the laser spectra and the upconverted HHG spectra withphase matching according to prior art Regimes I and II. In this regime,the central laser wavelength of driving laser is tuned from the near-IRtowards the mid-IR region of the spectrum. The laser bandwidths in themid-IR can be relatively broad to support readily both multi-cycle andfew-cycle driving pulses. Using longer laser wavelengths broaderphase-matched HHG bandwidths are generated that are also centered atmuch shorter HHG wavelength. The HHG bandwidths can extend from the EUVwell into the soft and hard X-ray regions of the spectrum.

FIG. 7B is a schematic spectral domain diagram illustrating the regionsof the laser spectra and the bright upconverted HHG spectra according topresent invention Regimes IIIA and IIIB. In these regimes, the centrallaser wavelength of driving laser is tuned from the near-IR towards theVIV-UV-VUV regions of the spectrum. That is, using shorter laserwavelengths and high laser intensity, bright, narrower HHG bandwidthsfrom atoms and low charged ions can be generated at longer HHGwavelength in the VUV-EUV regions. Also, using shorter laser wavelengthsand very high laser intensities can result in unexpectedly brightbroader HHG bandwidths from atoms and multiply charged ions at muchshorter HHG wavelength. These HHG bandwidths can extend from the EUVwell into the soft and hard X-ray regions of the spectrum.

FIG. 8 is a schematic spectral domain diagram illustrating output pulsespectral shapes of upconverted HHG light according to prior art RegimeII and the present invention, Regimes IIIA and IIIB. Using longer mid-IRlaser wavelengths, the phase matched HHG spectra exhibit supercontinuumstructure (prior art Regime II). This is due to the fact that the phasematching window within the driving pulse greatly reduces as we increasethe laser wavelength and the generation of an isolated HHG burst can bea norm. In the present invention the phase matching window increases forshorter laser wavelength. Furthermore, the resulting group velocitymismatch can be minimized. Thus either a broad comb of discrete brightharmonics can be generated in the spectral domain when a multi-cycle,shorter-wavelength UV laser pulse is used, or a supercontinuum spectrumcan be generated for a few-cycle driving pulse. The harmonic combproduced can have a very pure structure—the high order harmonics inRegime III can be well separated with fewer of them in the wholespectrum compared to Regime II. In addition, since the harmonic peaksgenerated by a shorter-wavelength UV laser are separated by a largerphoton energy spacing, isolation of a single harmonic or severalharmonics becomes more practical. For example, using a combination oflow-pass and high-pass thin film filters that simultaneously may rejectthe fundamental light, and selecting a specific laser intensity canisolate a single harmonic or several harmonics. Also, a proper selectionof nonlinear medium, thin film filters, and maximum phase matchingenergy can result in a similar harmonic selection.

FIG. 9A (Prior Art) is a schematic diagram illustrating the controlparameters, optimizing parameters, and resulting macroscopic HHG flux inconventional for prior art Regime I, and shows parameters and results atvarious wavelengths. Regime I generates photons in the EUV and in thesoft X-ray regions by using the same laser wavelength in the near-IRregion for which the resulting HHG flux from a single emitter does notvary significantly. Optimization here requires increasing the laserintensity as a control parameter, which reduces the usabledensity-length product (lower number of potential emitters) due toionization induced effects. Quantitatively, under some conditions thecoherence length can decrease rapidly to about 1-100 mm at very highlaser intensities and the optimal pressure can decrease to 1-5 torr. Theresulting non-phase-matched HHG flux drops rapidly at short wavelengths.

FIG. 9B (Prior Art) is a schematic diagram illustrating the controlparameters, optimizing parameters, and resulting macroscopic HHG flux inconventional for prior art Regime II, and shows parameters and resultsat various wavelengths. Regime II relies on increasing the laserwavelength towards the mid-IR spectral region as a control parameter.The resulting emission from a single-atom drops rapidly and may scale asI_(Laser) ^(−(7.5-9.5)) under optimal conditions depending on theselected bandwidth. Optimizing phase matching in this regime involvessimultaneous decrease in laser intensity. Furthermore, phase matchingconditions at shorter HHG wavelengths using longer driving wavelengthsfavors large optimal or phase matching density-length products (largenumber of potential emitters). For a comparison, under some conditionsthe optimal length can increase to about 0.5-1 cm and greater and theoptimal pressure can increase to about 1-40 atmospheres compensating forthe poor single-atom yield. As a result, the macroscopic phase-matchedHHG flux may vary slightly with wavelength.

FIG. 9C is a schematic diagram illustrating the control parameters,optimizing parameters, and resulting macroscopic HHG flux Regime IIIaccording to the present invention. The laser wavelength is decreasedtowards the VIS-UV-VUV region to generate bright harmonics in theVUV-EUV-X-ray regions of the spectrum. In contrast to Regime II, thisnew regime does not compromise in terms of a single-atom/single-ionemission. Here, the microscopic radiation increases rapidly as the laserwavelength decreases and may scale as fast as I_(Laser) ^(−7.5) for abandwidth of a single harmonic. To obtain bright phase matched oreffectively phase matched emission at higher photon energy the laserintensity has to be increased simultaneously with the decreasing thelaser wavelength. The pressure-length product here can be split intooptimal length and optimal pressure considering optimization. Inprinciple, the coherence length scales as L_(coh)μ1/(qI_(L)P) under someconsiderations, and can be allowed to decrease when decreasing I_(Laser)since the single-atom/single-ion yield is strong and shorter interactionlength may result in usable flux for applications. Furthermore, sincehigher photon energies are generated while higher ionization states areproduced the optimal pressure may increase. On one hand, the absorptiondrops at higher HHG photon energies. On the other hand, the absorptionof the ionized medium may also decrease during the time interval withinthe laser pulse when atomic and ionic species relevant to a specificabsorption disappear.

FIG. 10A (Prior Art) is a schematic time domain diagram illustrating theregions of the driving laser pulse durations and the HHG pulse durationswith phase matching in prior art Regimes I and II at variouswavelengths. The achievable pulse durations in prior art Regime I are onthe femtosecond-to-attosecond time scales. A smaller range of HHG outputpulse durations is achievable due to the narrower phase-matched HHGbandwidth available. The results achievable with prior art Regime II arein the femtosecond-to-attosecond-to-zeptosecond time scales. Theavailable HHG pulse duration range is larger due to the largerphase-matched HHG bandwidths. The driving laser pulses in Regime II arelonger than those in Regime I since the period of the laser pulse islonger for longer-wavelength mid-IR lasers—longer femtosecond timescales compared to shorter femtosecond time scales, respectively. Thetime scales of the laser pulses and the HHG pulses are shown on logscale.

FIG. 10B is a schematic time domain diagram illustrating the regions ofthe driving laser pulse durations and the HHG pulse durations with phasematching in prior art Regime I and Regime III according to the presentinvention at various wavelengths. FIG. 10B illustrates achievable HHGpulse durations with devices according to the present invention. A broadrange of pulse durations similar to that in Regime II can beobtained—femtosecond-to-attosecond-to-zeptosecond time scales. This isbecause the bright emission from ions can extend the available brightcoherent HHG bandwidth. In contrast, the driving pulse durations cover abroader range compared to Regimes I and II. First, using shorter UV-VUVlaser wavelengths means that the driving pulse may have a pulse durationin the attosecond time scale. Second, phase matching and greatly reducedphase mismatch can allow longer pulse durations to be used—much greaterthan 10-100 fs (significantly longer durations compared to the durationof the cycle at these wavelengths). The reason for this is that theefficient VUV-UV-VIS driven HHG can tolerate much larger ionizationlevels that increase with every additional laser cycle.

FIGS. 11A-D are time domain diagrams illustrating input laser pulseshapes and output HHG pulse shapes according to prior art Regime II.FIG. 11A depicts a single driving ultrashort pulse input. FIG. 11Bindicates that a more complex series of pulses, for example, FEL pulsesor shaped pulses, consisting of several ultrashort micro-pulses within along, macro-pulse, can be used. FIG. 11C shows that a series of pulsesmay be generated by the driving pulse of either 11A or 11B. Note thateach HHG pulse is chirped to a relatively long pulse duration comparedto its transform limit that can be reached after compression. Since thephase matching window narrows as we increase the laser wavelength, FIG.11D shows that a single output pulse may be easily achieved.

FIGS. 12A-D are time domain diagrams illustrating input laser pulseshapes and output HHG pulse shapes according Regime III of the currentinvention. Similar input laser pulse shapes can be used. In FIG. 12A,the input pulse may have either small or relatively large numbers ofcycles since the phase matching window can be longer and the resultinggroup velocity can be mitigated. The complex series of pulses in FIG.12B may be closer in micro or macro separation since higher ionizationlevel can be tolerated. A proper design of the micro and macro pulsesmay lead to an effective increase of the repetition rate of the laserwith the purpose to increase the number of HHG photons per sec.Depending on the number of driving laser cycles, the output HHG pulsesmay be a train of HHG pulses as in FIG. 12C, or an isolated HHG pulse asin FIG. 12D. Each HHG pulse is chirped to a shorter pulse durationcompared to Regime II (shown in FIGS. 11A-D). Again, since the phasematching window is longer and the driving pulses are less susceptible togroup velocity walk off effects, the output HHG flux can be increasedadditionally by increasing the number of laser cycles/HHG burst.

FIG. 13 is a time domain diagram presenting the generated, chirped, HHGpulse durations and the minimum HHG pulse durations if the pulses arecompressed for prior art Regime II and Regimes IIIA and IIIB accordingto the present invention at various wavelengths.

The time domain diagram in FIG. 13 shows a generalized picture of thepulse durations of the generated HHG pulses as a function of the drivinglaser wavelength for Regime II (right-hand side of figure) compared toRegime III, the present invention (left-hand side of figure). The HHGchirp (the group delay dispersion, GDD) scales as:IGDDIμT_(LASER)/U_(p)where T_(LASER) is the laser cycle duration and the U_(p)μI_(L)I_(L) ².Using longer wavelength the chirp decreases approximately asIGDDI˜1/I_(L) ^(0.7) since the phase matching cutoffs scaleapproximately as hnμI_(L) ^(1.7). As the available phase matchedbandwidths increase almost quadratically the HHG pulses generateddirectly from the medium have very long pulse durations in the ˜1-10 fsrange for mid-IR drivers while their transform limited duration aftercompression can be as short as ˜10 attosecond-500 zeptosecond,respectively. In contrast in Regime III, the chirp can be smaller whencompared for the same bandwidth at the same photon energy. Thus thegenerated HHG pulses can be closer to their transform limit. In somecases this may eliminate the need for an HHG compressor that in generalis lossy or may reduce the attenuation of the HHG intensity when a smallcompression is needed. As an example, a 1.3 mm laser can generated phasematched HHG up to ˜300 eV in He with HHG pulse durations of >600attoseconds. A 0.27 mm laser can generate the same bright HHG cutoff.While the transform limited pulse duration in both cases is almost thesame for the same bandwidth, the generated HHG pulses in Regime III have4-5 smaller chirp compared to those in Regime II and can be as shortas >125 attosecond before compression. If the laser intensity in RegimeIII is increased further, the chirp can be decreased to even smallervalues, inversely proportionally to the laser intensity.

FIG. 14 is a plot of HHG phase matching limits for prior art Regime II(right-hand side of figure) and efficient HHG upconversion photonenergies in Regimes IIIA and IIIB according to the present invention(left-hand side of figure). FIG. 14 illustrates that by using longerlaser wavelength and lightly ionized atoms, bright coherent soft andhard X-rays can be generated (prior art, Regime II). According to thepresent invention (Regime III), by using shorter laser wavelengths andfully- or multiply-charged ions, bright soft and hard X-rays can begenerated. While Regime IIIA may extend the efficient upconversionmostly in the VUV-EUV region, Regime IIIB can extend the emission wellinto the X-ray region. The two approaches in Regime II and Regime IIIare complimentary to each other. The first one generates preferablycoherent, supercontinuum, X-ray harmonics ideal for spectroscopic typeof applications, while the second generates preferably coherent,discrete, X-ray harmonics ideal for imaging type of applications.

FIG. 15A is a plot showing example experimental data demonstratingspectrally selected bright discrete harmonics in the X-ray region of thespectrum up to the water window in Regime IIIB according to the presentinvention (linear intensity scale). Experimentally observed UV-drivenHHG extends up to the water window at 280 eV, for laser intensities of>6×10¹⁵ W/cm². FIG. 15A—linear intensity scale.

FIG. 15B is a plot showing example experimental data presenting brightdiscrete harmonics spanning the VUV-EUV-X-ray spectral regions (logintensity scale). The plot shows all the generated VUV-EUV-X-raydiscrete narrowband harmonics except for the 3rd harmonic. Calculationshow that the emission above 30 eV must originate from multiply chargedions (Ar⁺—Ar⁶⁺).

FIG. 16 is a plot showing example experimental data demonstratingultra-bright discrete harmonics in the EUV spectral region in RegimeIIIA according to the present invention. FIG. 17 is a plot showing anexample calculation of the critical ionization level of a gas versuslaser wavelength for Regime IIIA according to the present invention,with and without spatial averaging.

In Regime IIIA, when the laser wavelength is decreased and the emissionis predominantly from neutral atoms, full phase matching limits decreasefrom the EUV towards the VUV region. Here, low charged ions compensateadditionally for a larger plasma dispersion boosting the macroscopic HHGflux. The conversion efficiency may approach 10⁻⁴ for a 0.4 mm driver,and 10⁻³ for a 0.27 mm driver. In some cases, for shorter laserwavelengths and noble gas atoms with larger number of electrons, such asAr, the critical ionization above which phase matching is not possiblemay approach 100% for a 0.2 mm driver as shown in FIG. 17. Usingaveraging of the index of refraction across the laser beam profile showsthat the critical ionization that can be tolerated may be increase toabout 400% in this case. This means that efficient upconversion may bepossible from multiply charged ions (Ar⁺—Ar⁴⁺) and increasing the laserintensity further will increase the phase mismatch slowly. For atomswith even larger number of electrons like Kr and Xe, which have higherneutral atom and ion indices of refraction (linear and nonlinearindices), the critical ionization without spatial averaging can be above100% meaning that full phase matching is possible in ions. Using spatialaveraging, the estimated critical ionization can reach fractional levelsof ionization corresponding to much higher charged states.

FIG. 18A is a plot showing an example calculation of the expected HHGemission from a single atom versus driving laser wavelength for RegimeIIIA according to the present invention. The favorable yield from asingle emitter is predicted to scale as I_(L) ^(−7.5) for a bandwidth ofa single harmonic and for the conditions in FIG. 18B. This yield isexpected to scale more strongly at higher laser intensities.

FIG. 18B is a plot showing an example calculation of the expectedefficient macroscopic harmonic intensity versus corresponding HHG photonenergy and driving laser wavelength for Regime IIIA according to thepresent invention. Using the method shown in FIG. 21, one can estimatethe increase in HHG flux as the laser wavelength is decreased. Theillustrated example covers Regime III A where the absorption length isrelatively short and full phase matching is possible. This estimatesexplain the enhanced experimental HHG intensities presented in FIG. 16.Using best estimates of the indices of refraction of the medium one canexpect >600 enhancement in macroscopic absorption limited HHG intensityfor a 0.27 mm driver and >50 enhancement for a 0.4 mm driver, comparedto a phase-matched emission for a 0.8 mm driver. The total enhancementis due to an enhanced, favorable macroscopic phase matching, as well asto a favorable microscopic yield.

FIGS. 19A, B and C illustrate embodiments where isolated singleharmonics or few harmonics are generated using a combination of low-passfilters and high-pass filters. This approach is attractive fromapplication point of view since it may reduce the bandwidth requirementor may completely eliminate the need for multilayer VUV-EUV-X-raymirrors that introduce significant loss to select a single harmonic.Using a few-cycle driving pulse in principle can also result in aspectral supercontinuum corresponding to an isolated VUV, EUV or X-raypulse.

FIG. 19A is a plot showing example experimental data demonstrating beamprofile and a bright narrow-band EUV harmonic isolated by low-pass andhigh pass thin film filters and by using a specific laser intensity.FIG. 19A demonstrates an isolated, single, 13 nm EUV harmonic (97 eV)with narrow linewidth of λ/Δλ˜250 or >350 meV. The inset shows the 13 nmHHG beam. The adjacent harmonic orders were filtered out using Rh+Si+Befilters of selected thicknesses.

FIG. 19B is a plot showing example experimental data demonstrating twoisolated, EUV harmonics around 13 nm with narrow linewidths. Theadjacent harmonic orders were filtered out using Rh+Be filters ofselected thicknesses.

FIG. 19C is a plot showing example experimental data demonstrating abright narrow-band VUV harmonic isolated by low-pass and high pass thinfilm filters and by using a specific laser intensity. The plotdemonstrates isolated, single, VUV harmonics around 54 nm (23 eV) withnarrow linewidth of λ/Δλ˜300. The adjacent harmonic orders were filteredout using an Sn filter of selected thickness.

FIG. 20A is a plot showing an example of an extended temporal window ofefficient upconversion over a large number of driving laser cycles forRegime III according to the present invention.

FIG. 20B (Prior Art) is a plot showing an example of a narrow temporalwindow of upconversion for prior art Regime I.

Calculated ratio of the dynamic phase mismatch due to plasma andatoms/ions for multi-cycle 0.27 μm and 0.8 μm driving lasers arecompared in FIGS. 20A and 20B respectively. In these plots, thehighlighted area shows the time window during which HHG emission can beas bright or brighter than the HHG in He driven by 0.8 μm lasersreaching the same photon energy. This time window is larger forUV-driven HHG than for 0.8 μm driven HHG. The dotted lines plot thecontributions of neutral atoms alone, while the bold solid lines includeboth atomic and ionic contributions to the linear and nonlinear indices.The regions enclosed by the dashed lines give the range of valuespossible: the high values of this ratio exclude the nonlinear indices ofrefraction of atoms and ions, while the lower values plot a best casescenario, assuming the UV indices of the ions is comparable to that ofneutral atoms. The dashed black horizontal lines bracket the same ratiosof plasma mismatch and atomic/ionic mismatch.

FIG. 21 is a flow diagram illustrating steps in the process of globaloptimization of HHG with phase matching or effective phase matchingaccording to the present invention, and steps in selecting the optimalparameters for most efficient HHG at a desired HHG wavelength.

In step 200, λ_(L) of the driving pulse is chosen. In step X, nonlinearmedium is selected. In step 202, the index of refraction of mediumconsisting of neutral targets and of multiply-charged ionic targets fora range of λ_(L) is computed.

The same step now is performed 206 for the nonlinear indices of neutraland ionic targets versus time dependent laser intensity and versus laserwavelength. In the following steps 208, 210, 212, 214, the geometry ofthe device is chosen. In step 216, the (effective) phase matching limitsas a function of λ_(L) are evaluated based on the total index ofrefraction of the driving laser light, and that of the generated light,for a specific nonlinear medium and a specific laser beam geometry. Thesame step may optimize the HHG photon energies for which we achieve thesame number of HHG photons, i.e. estimating a finite coherence length asa function of the driving laser wavelength that results in a desirednumber of photons. In step 218, the optimal laser pulse parameters(laser intensity, pulse duration, pulse shape, etc.) are determined.Since the total index of refraction of the neutral and ionic medium isdependent on the laser intensity the indices of refraction of thenonlinear medium may need to be recalculated iteratively. In step 220,the HHG flux from a single emitter as a function of λ_(L) is determined.The generated HHG light in the nonlinear medium can be absorbed by themedium. Thus the monotonic growth of the HHG signal may saturate whichsets an optimal density-length product (or number of potential emitters)that can be used. The density-length product is determined by theproperties of nonlinear medium. This is the case if the coherence lengthis greater than the absorption length. If the coherence length issignificantly shorter than the absorption length and results is adesired number of photon the selection of the pressure is optimized withrespect to an optimal coherence length. For geometries where (effective)phase matching is pressure dependent (for example, waveguidegeometries), first optimal pressure is determined and then optimallength, set by the same density-length product. The pressureoptimization here may involve minimizing the phase mismatch to achieve adesired L_(coh). Step 228 combines the microscopic, single emitter HHGflux, and the optimization of macroscopic nonlinear medium parametersand laser parameters to get the macroscopic phase-matched HHG flux ornumber of photons as a function of laser wavelength. Flux can giverelative brightness of phase-matched HHG of a desired wavelength forvarious nonlinear medium, laser wavelengths, etc. Determining themaximum flux or a desired number of photons at a desired HHG wavelengthallows proper selection of optimal laser wavelength, optimal nonlinearmedium, optimal driving laser pulse parameters, optimal density lengthproduct.

While the exemplary preferred embodiments of the present invention aredescribed herein with particularity, those skilled in the art willappreciate various changes, additions, and applications other than thosespecifically mentioned, which are within the spirit of this invention.

FIG. 22 is experimental data demonstrating optimization of bright X-rayHHG extending up the water window at >280 eV versus pressure for a laserintensity I_(L)>6×10¹⁵ W/cm².

Using a 0.27 μm laser and very high laser intensity I_(L)>6×10¹⁵ W/cm²,bright emission from multiply charged Ar ions can extend up to the waterwindow at >280 eV (Regime III B). Note that the optimal pressureincreases to above 400 torr for the highest X-ray harmonics. Incomparison, using a 0.8 μm laser in Regime I, the optimal pressure for anon-phase matched HHG from multiply charged Ar ions is 2 orders ofmagnitude lower—˜1-5 torr. Since the coherence length in Regime IIIincreases much slowly with harmonic order and the same photon energycorresponds to a lower harmonic order compared to Regime II, higheroptimal pressures can be allowed. As mentioned above, a shortercoherence length in Regime III can lead to a bright HHG emission alsobecause of strong emission from a single atom. Furthermore, as highlycharged states are generated during the laser pulse for a specifictemporal window the absorption of the medium related to the fullyionized states can be reduced.

What is claimed is:
 1. A method of generating coherent emission withinthe VUV, EUV, and X-ray region of the spectrum comprising the steps of:(a) providing a nonlinear medium comprising a gas for high harmonicgeneration (HHG) of laser light; (b) selecting the pressure and lengthof the nonlinear medium; (c) generating a laser driving pulse having aselected wavelength within the VUV-UV-VIS region of the spectrum and aselected peak intensity above 5×10¹⁴ W/cm²; (d) focusing the drivingpulse into the nonlinear medium to cause HHG upconversion resulting in agenerated pulse within a selected range within the VUV, EUV, and X-rayregion of the spectrum; and (e) substantially phase matching the drivingpulse and the generated pulse within the nonlinear medium sufficientlyto result in a conversion efficiency from laser energy per pulse intoHHG energy per pulse of at least 10⁻⁹.
 2. The method of claim 1 whereinstep (e) phase matches the driving pulse and the generated pulse, andwherein phase matching is accomplished by balancing the combineddispersion of neutral atoms and low-charged ions, and the plasmadispersion (Regime III A).
 3. The method of claim 2 wherein theionization of the nonlinear medium is at least about 10% (Regime III Aand B).
 4. The method of claim 1 wherein the coherence length is atleast about 0.1 cm.
 5. The method of claim 1 wherein step (e)effectively phase matches the driving pulse and the generated pulse, andwherein effective phase matching is accomplished by partially balancingthe combined dispersion of neutral atoms and multiply-charged ions, andthe plasma dispersion (Regime III B).
 6. The method of claim 5 whereinthe ionization of the nonlinear medium exceeds 100% fractionalionization (Regime III B).
 7. The method of claim 5 wherein thecoherence length is at least about 0.1 μm.
 8. The method of claim 1wherein providing a nonlinear medium provides a gas phase medium thatcan be ionized.
 9. The method in claim 1 where the nonlinear medium isconfined in a gas cell.
 10. The method of claim 9 further including thestep of maintaining a high-pressure gas region within the gas cell wherethe laser is focused to high intensity to drive the nonlinearinteraction, and wherein gas pressure drops rapidly upon exiting the gascell to limit re-absorption of the light generated.
 11. The method inclaim 1 where the nonlinear medium is confined in a waveguide.
 12. Themethod of claim 11 further including the step of maintaining ahigh-pressure gas region within the waveguide where the laser is focusedto high intensity to drive the nonlinear interaction, and wherein gaspressure drops rapidly upon exiting the waveguide to limit re-absorptionof the light generated.
 13. The method of claim 11 further including thestep of keeping the driving laser at high intensity simultaneously withkeeping the nonlinear gas medium at high intensity.
 14. The method inclaim 1 where the laser beam self-confines in space and its pulseduration self-confines in time during the propagation in the nonlinearmedium so that there is no substantial decrease in laser intensity. 15.The method of claim 1 wherein the driving laser pulse has a wavelengthbelow 0.7 μm.
 16. The method of claim 15 wherein the driving laser pulsehas a wavelength of about 0.4 μm or shorter.
 17. The method of claim 16wherein the driving laser pulse has a wavelength of about 0.27 μm orshorter.
 18. The method of claim 1 wherein step 1(c) of generates adriving pulse having a duration of below 350 femtoseconds and an energyof between 100 μJ and 1 J and repetition rate between 0.1 Hz and 1 MHz.19. The method of claim 1 resulting in a generated pulse having aduration within the femtosecond to zeptosecond range.
 20. The method ofclaim 1 resulting in a generated comb of narrow-band harmonics.
 21. Themethod of claim 20 resulting in a generated isolated harmonic.
 22. Amethod for global flux optimization of coherent harmonic emission atdesired wavelength comprising the steps of: (a) evaluating a total indexof refraction of neutral and ionic species of a medium as a function ofthe wavelength of a driving laser and evaluating an index of refractionof the corresponding harmonic emission; (b) evaluating a total index ofrefraction of neutral and ionic species of a medium at a laserwavelength as a function of the interaction time between the medium andthe driving laser pulse; (c) evaluating full phase matching andeffective phase matching limits of HHG as a function of the drivinglaser wavelength; (d) determining optimal laser parameters as a functionof the a driving laser wavelength; (e) evaluating the flux from a singleemitter as a function of the driving laser parameters under full phasematching and effective phase matching conditions; (f) evaluating theoptimal density-length product of the medium as a function of thedriving laser wavelength; (g) combining steps (a)-(f) and calculatingthe macroscopic phase matched HHG emission and effectively phase-matchedHHG emission as a function of the wavelengths of the generated light andthe driving laser light; and (h) finding a global maximum of flux at adesired HHG wavelength and selecting an optimal driving laserwavelength, optimal laser pulse parameters, optimal spatial and temporallaser pulse shape, optimal type of nonlinear medium and parameters ofthe nonlinear medium.
 23. The method in claim 22 where globaloptimization is performed to adjust the bandwidth of individualharmonics while optimizing the flux.
 24. The method in claim 22 whereglobal optimization is performed for harmonics with wavelengths around13 nm or shorter.
 25. The method in claim 24 where global optimizationis performed for harmonics with wavelengths around 6 nm or shorter. 26.The method of implementing high-harmonic generation with parametersconsistent with those obtained in the method of claim 22.